### Show that the logarithmic mean area Am of a cylinder of unit length, internal radius r1, external radius r2, thermal conductivity k, whose inside and outside surface temperature are t1 and t2 respectively are given by: A_m=(2p(r_2-r_1))/(l_n r_2/r_1 ) (6 marks) b) Wet steam at 20 bar is carried in a steel steam mains of outside diameter 140mm, thickness 6mm and length 8m. The mains is insulated with an inner layer of diatomaceous earth 38mm, and an outer layer of magnesia 30mm thick. The inside heat transfer coefficient is 8.5 W/m2K and that of the outside surface lagging is 18W/m2K. The thermal conductivities of the diatomaceous earth, magnesia and steel are 0.09, 0.06 and 48W/mK, respectively. Determine: The rate of heat loss. The temperature of the outside surface of the lagging.

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a) Explain the following terms:
Geometrical similarity
Dynamic similarity (3 marks)

b) Show from the first principles the requirements for dynamic similarity between two fluid motions when considering:
Viscous resistance
Wave resistance (6 marks)

The air resistance R of a supersonic plane during flight, is a function of its length L, velocity V, air dynamic velocity µ, air density ?, and bulk modulus K. Show that the air resistance R is given by:

R=pl^2 V^2Ø{µ/pvl ,k/(pv^2 )}

where Ø means a “function of”. (11 marks)

a) i) Define the following terms with reference to fluid flow:
Critical velocity
Reynold’s number
ii) Show that the flow rate Q, of a fluid of dynamic viscosity ?, flowing under laminar conditions through a horizontal circular pipe of diameter ‘d’, length ‘L’, with a mean velocity V, when the pressure difference between the ends is P, is given by the expression:

Q=(ppd^4)/128?L
Hence show that the pressure difference p can be given by the expression

P=32?lv/d^2 (121/2 marks)

b) Oil is pumped through a pipe 120mm diameter and 900m long. The pressure difference between the ends is 420KN/m2. The dynamic viscosity of the oil is 1.42 N-S/m2 and the relative density is 0.9.

Show that the flow is viscous if Reynold’s number is 2100.
Calculate the electric power of the motor required if the mechanical efficiency between the pump and the electric motor is 80%.

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